Let $\{u_1,..,u_k\}$ be orthonormal.
Then that means $\langle u_i,u_j\rangle = \begin{cases}1 & ,i = j\\0 & , else\end{cases}$
"What happens when $\langle u_k,u_k\rangle\neq 1$?"
Well, then it's (obviously) not orthonormal anymore?
No idea. What are the implications beyond that?
As was pointed out int he comments, such a basis would be called an orthogonal basis - so long as $\langle u_k, u_k \rangle \neq 0$ for all $k$. It doesn't have all the nice properties that an orthonormal basis would though; with an orthonormal basis it's not hard to prove that one can compute the inner product of two vectors by summing the products of the coordinates:
$$ \langle a, b \rangle = \sum_{i=1}^n a_ib_i,$$ where $a_i, b_i \in \mathbb{R}$ denote the coordinates of $a$ resp. $b$. With an orthogonal basis (as with all others), this doesn't hold in general. Any orthogonal basis can be easily 'transformed' into an orthonormal one though, by some simple rescaling of each vector (though any basis can be transformed into an orthonormal one by a Gram-Schmidt process, so it's not a very special property or anything).